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This paper investigates an L1 adaptive fault-tolerant control scheme for nonlinear systems with input constraint, external disturbances, and multiple faults, which include actuator faults and sensor faults. Faults and input constraint are important factors that affect the stability and performance of a control system. Actuators and sensors are the most vulnerable components, with the former receiving more attention in comparison. In this paper, sensor faults are first transformed into pseudo-actuator faults through the augmented matrix approach, which facilitates their handling together with actuator faults. Saturation constraints on the control signal are not conducive to the design of the controller. The conversion of an input-saturated function to a time-varying linear system is completed based on function approximation and Lagrange's mean value theorem. Moreover, a nonlinear system with unknown input gain and uncertainties is constructed using these methods. Next, an L1 adaptive fault-tolerant controller is designed to cope with uncertainties, including system uncertainties, external disturbances, faults, and approximation errors. In the L1 adaptive controller, the online estimation of the time-varying parameters allows for updating of the system state, while the combination of the two is transmitted to the control law such that it can compensate for the effects of the uncertainties. The stability and performance boundaries are further derived using the Lyapunov theory and the L1 reference system. Finally, simulations are carried out to demonstrate the effectiveness of the proposed controller. [ABSTRACT FROM AUTHOR]

To address the problem of low accuracy and efficiency in trajectory planning algorithms for interceptors facing multiple constraints during the midcourse guidance phase, an improved trajectory convex programming method based on the lateral distance domain is proposed. This algorithm can achieve fast trajectory planning, reduce the approximation error of the planned trajectory, and improve the accuracy of trajectory guidance. First, the concept of lateral distance domain is proposed, and the motion model of the midcourse guidance segment in the interceptor is converted from the time domain to the lateral distance domain. Second, the motion model and multiple constraints are convexly and discretely transformed, and the discrete trajectory convex model is established in the lateral distance domain. Third, the deep reinforcement learning algorithm is used to learn and train the initial solution of trajectory convex programming, and a high-quality initial solution trajectory is obtained. Finally, a dynamic adjustment method based on the distribution of approximate solution errors is designed to achieve efficient dynamic adjustment of grid points in iterative solving. The simulation experiments show that the improved trajectory convex programming algorithm proposed in this paper not only improves the accuracy and efficiency of the algorithm but also has good optimization performance. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study a sequence of linear and positive operators which was introduced by A. Lupas. First, we obtain estimate of moments of the operators and then prove a basic convergence theorem for simultaneous approximation. Further, we find error in approximation in terms of modulus of continuity of function. Finally, we establish a Voronovskaja asymptotic formula for linear combination of the above operators. [ABSTRACT FROM AUTHOR]

This paper studies an adaptive fault tolerant control (AFTC) scheme for a continuum robot subjected to unknown actuator faults, dynamics uncertainties, unknown disturbances, and prescribed performance. Specifically, to deal with uncertainties, a function approximation technique (FAT) is employed to evaluate the unknown actuator faults and uncertain dynamics of the continuum robot. Then, a nonlinear disturbance observer (DO) is developed to estimate the unknown compounded disturbance, which contains the unknown disturbances and approximation errors of the FAT. Furthermore, the prescribed error bound is treated as a time-varying constraint, and the controller design method is based on an asymmetric barrier Lyapunov function (BLF), which is operated to strictly ensure the steady-state and transient performance of the continuum robot. Afterwards, the simulation results validate the effectiveness of the proposed AFTC in dealing with the unknown actuator faults, uncertainties, unknown disturbances, and prescribed performance. Finally, the effectiveness of the proposed AFTC scheme is verified through experiments. [ABSTRACT FROM AUTHOR]

This paper proposes a dead-reckoning (DR) method for vehicles using Lie theory. This approach treats the pose (position and attitude) and velocity of the vehicle as elements of the Lie group SE 2 (3) and follows the computations based on Lie theory. Previously employed DR methods, which have been widely used, suffer from cumulative errors over time due to inaccuracies in the calculated changes from velocity during the motion of the vehicle or small errors in modeling assumptions. Consequently, this results in significant discrepancies between the estimated and actual positions over time. However, by treating the pose and velocity of the vehicle as elements of the Lie group, the proposed method allows for accurate solutions without the errors introduced by linearization. The incremental updates for pose and velocity in the DR computation are represented in the Lie algebra. Experimental results confirm that the proposed method improves the accuracy of DR. In particular, as the motion prediction time interval of the vehicle increases, the proposed method demonstrates a more pronounced improvement in positional accuracy. [ABSTRACT FROM AUTHOR]

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed. [ABSTRACT FROM AUTHOR]

This paper presents general theoretical studies on asymptotic convergence rate (ACR) for finite dimensional optimization. Given the continuous problem function and discrete time stochastic optimization process, the ACR is the optimal constant for control of the asymptotic behaviour of the expected approximation errors. Under general assumptions, condition ACR < 1 implies the linear behaviour of the expected time of hitting the ε - optimal sublevel set with ε → 0 + and determines the upper bound for the convergence rate of the trajectories of the process. This paper provides general characterization of ACR and, in particular, shows that some algorithms cannot converge linearly fast for any nontrivial continuous optimization problem. The relation between asymptotic convergence rate in the objective space and asymptotic convergence rate in the search space is provided. Examples and numerical simulations with use of a (1+1) self-adaptive evolution strategy and other algorithms are presented. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive bearing rigid formation control strategy for a class of nonlinear system with unknown dead-zone inputs and external disturbance is proposed. Firstly, the I-Type fuzzy system is used to approximate the unknown nonlinear dynamics of the formation model, and the approximation errors and unknown external disturbance are eliminated by the parameter adaptive estimation. Furthermore, the adaptive dynamic estimation algorithm is utilized to estimate and compensate the unknown dead-zone parameters, effectively suppressing the impact of dead-zone on formation system performance. Finally, the stability of the formation system is proved based on LaSalle's invariance principle, and the effectiveness of the algorithm is verified by simulation results. [ABSTRACT FROM AUTHOR]

Large-scale symmetric and asymmetric matrices have emerged in predicting the relationship between genes and diseases. The emergence of large-scale matrices increases the computational complexity of the problem. Therefore, using low-rank matrices instead of original symmetric and asymmetric matrices can greatly reduce computational complexity. In this paper, we propose an approximation conjugate gradient method for solving the low-rank matrix recovery problem, i.e., the low-rank matrix is obtained to replace the original symmetric and asymmetric matrices such that the approximation error is the smallest. The conjugate gradient search direction is given through matrix addition and matrix multiplication. The new conjugate gradient update parameter is given by the F-norm of matrix and the trace inner product of matrices. The conjugate gradient generated by the algorithm avoids SVD decomposition. The backtracking linear search is used so that the approximation conjugate gradient direction is computed only once, which ensures that the objective function decreases monotonically. The global convergence and local superlinear convergence of the algorithm are given. The numerical results are reported and show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we develop a method for the estimation of airgap spatial variation from the harmonics that appear in the stator's current under static eccentricity conditions. We move beyond the detection of the eccentricity fault and endeavour to quantify the severity of this fault. To achieve this, we formulate the problem as a mathematical inverse problem and attempt to solve it in the Bayesian framework using the so-called Bayesian approximation error approach. The solution takes the form of an algorithm capable of estimating the parameter(s) that characterize the spatial variation of the airgap length from noisy and limited data. Furthermore, it can quantify the uncertainty associated with the results as a byproduct of the Bayesian analysis approach employed in this paper. The results demonstrate that with each iteration of the algorithm, estimation accuracy improves, and the uncertainty of the estimated results reduces. • Bayesian approximation error analysis shows promise in monitoring electrical machines with eccentric rotors. • The paper applies Bayesian inversion for online estimation of airgap length in induction machines with eccentric rotors. • The study estimates static eccentricity ratio from limited data on harmonics of the machine line currents. • This paper gives an analytical method for calculating all inductances of induction machines under static eccentricity. [ABSTRACT FROM AUTHOR]

This paper presents application and tuning of the equilibrated residual method (ERM) of a posteriori error estimation for coupled electromechanical problems of direct, inverse and general piezoelectricity. In these three cases, either electric potential is induced by strains or strains appear due to the applied electric potential or both phenomena occur simultaneously. The mentioned ERM is assigned for the assessment of modeling and approximation errors of the numerical finite element solution. Such error values usually serve as indication for adaptive hierarchical modeling and adaptive mesh changes within thin and/or solid piezoelectric members so as to obtain the solution of assumed accuracy. [ABSTRACT FROM AUTHOR]

Bayesian optimization algorithms are widely used for solving problems with a high computational complexity in terms of objective function evaluation. The efficiency of Bayesian optimization is strongly dependent on the quality of the surrogate models of an objective function, which are built and refined at each iteration. The quality of surrogate models, and hence the performance of an optimization algorithm, can be greatly improved by selecting the appropriate hyperparameter values of the approximation algorithm. The common approach to finding good hyperparameter values for each iteration of Bayesian optimization is to build surrogate models with different hyperparameter values and choose the best one based on some estimation of the approximation error, for example, a cross-validation score. Building multiple surrogate models for each iteration of Bayesian optimization is computationally demanding and significantly increases the time required to solve an optimization problem. This paper suggests a new approach, called exploratory landscape validation, to find good hyperparameter values with less computational effort. Exploratory landscape validation metrics can be used to predict the best hyperparameter values, which can improve both the quality of the solutions found by Bayesian optimization and the time needed to solve problems. [ABSTRACT FROM AUTHOR]

The paper provides a precise error estimate for an asymptotic expansion of a certain stochastic control problem related to relative entropy minimization. In particular, it is shown that the expansion error depends on the regularity of functionals on path space. An efficient numerical scheme based on a weak approximation with Monte Carlo simulation is employed to implement the asymptotic expansion in multidimensional settings. Throughout numerical experiments, it is confirmed that the approximation error of the proposed scheme is consistent with the theoretical rate of convergence. [ABSTRACT FROM AUTHOR]

When preparing a pure state with a quantum circuit, there is an unavoidable approximation error due to the compilation error in fault-tolerant implementation. A recently proposed approach called probabilistic state synthesis, where the circuit is probabilistically sampled, is able to reduce the approximation error compared to conventional deterministic synthesis. In this paper, we demonstrate that the optimal probabilistic synthesis quadratically reduces the approximation error. Moreover, we show that a deterministic synthesis algorithm can be efficiently converted into a probabilistic one that achieves this quadratic error reduction. We also numerically demonstrate how this conversion reduces the T-count and analytically prove that this conversion halves an information-theoretic lower bound on the circuit size. In order to derive these results, we prove general theorems about the optimal convex approximation of a quantum state. Furthermore, we demonstrate that this theorem can be used to analyze an entanglement measure. [ABSTRACT FROM AUTHOR]

This paper provides a convergence and stability analysis of the incremental value iteration algorithm under the influence of various errors. Incremental control is firstly used to linearize the continuous-time nonlinear system, recursive least squares (RLS) identification is then introduced to identify the incremental model online. Based on the incremental model, the value iteration algorithm is used to design an optimal adaptive controller, with an analytical optimal control law. Moreover, the convergence of the developed incremental value iteration algorithm is proved. The stability of the controller is analyzed using Lyapunov stability theory. Finally, a flight control simulation verifies the robustness of the controller to various initial conditions, as well as adaptation to actuator faults. [ABSTRACT FROM AUTHOR]

Let ω be a positive, Lebesgue integrable and exponential decay function defined on an infinite interval I and let L1,ω(I) be the space of weighted Lebesgue integrable functions on I. In this paper, we give the relations of the best one-sided approximation and the optimal Hermite–Fejér interpolation by the set of algebraic polynomials of degree not exceeding a given number for the smooth function classes BC2r,+(I), r ∈ ℕ+, in the metric of the space L1,ω(I), and prove that the Hermite–Fejér interpolation based on the set of the zeros of some orthogonal polynomials is optimal in L1,ω(I). In addition, we show that the approximation error of the optimal Hermite–Fejér interpolation and quadrature errors of the weighted Gaussian quadrature formula are equal, and give the exact constant of the errors. [ABSTRACT FROM AUTHOR]

An adaptive neural network energy-based control approach is proposed in this paper to address the issue of stabilizing a partial space elevator during cargo transportation. The method handles the challenges presented by external disturbances and model uncertainties. The dynamic model of the partial space elevator, with a typical multi-input and multi-output underactuated system, is presented. A control scheme using energy-based techniques is proposed to guarantee the stable configuration of the system, and auxiliary kinematic vector variables are designed to incorporate the uncontrollable variables to constitute controllable state subsets. Under these frameworks, adaptive control schemes are formulated using multi-layer neural networks to mitigate model uncertainty. Robust compensators are designed to effectively counteract the unknown boundaries of external disturbances and approximation errors. The stability of the closed control system is rigorously demonstrated through the application of the Lyapunov function and LaSalle's invariance principle. Numerical simulations demonstrate that the proposed control method is highly effective in suppressing oscillations during the transfer period of a partial space elevator. [ABSTRACT FROM AUTHOR]

Recently, Cheval-Garabédian et al. (2023) published a paper in Ore Geology Reviews entitled "The La Lucette Sb-Au-(W) vein-deposit (Armorican Massif, France): new time and genetic constraints to decipher the 310–295 Ma Sb-Au metallogenic peak in the Variscan Belt." Studies on Sb mineralisation are highly welcomed, especially in France, where there is a lack of updated data hindering a comprehensive understanding of this mineral system. The paper provides new mineralogical, fluid inclusion and geochronological data from historical collection samples of La Lucette mine, a mining site no more accessible since at least 40 years (Serment, 1978). These data, together with structural data from literature, lead the authors to build a metallogenic model related to dextral shear zone during the Carboniferous-Permian transition. The authors further argue for a single and unique 310–295 Ma Sb-Au metallogenic peak in the European Variscan Belt, a late Variscan hydrothermal event coeval with post-thickening tectonics (Munoz et al., 1992; Bouchot et al., 1997, 2005). This assertion is not correct. In this comment, we highlight that (i) several errors and approximations question the validity of their model, and (ii) the authors omit certain information that contradicts their metallogenic model. In our view, their data do not fully support the existence of a generalised and unique Late Carboniferous Sb-Au metallogenic peak in the French Variscan Belt. Furthermore, the idea defended by the authors that ore deposits occurred during dextral strike-slip tectonics that affected the area may better support a possible long-lived history during Palaeozoic times. [ABSTRACT FROM AUTHOR]

Reconstructed temperature field will provide critical input for monitoring the aerodynamic heating on the thermal protection system (TPS). In this paper, a novel data-driven method is proposed that allows real-time reconstruction of the 3D transient non-uniform temperature field in a TPS ceramic tile. The measured temperature is obtained by embedding sparse sensors at different layers of the tile. The spatial feature bases and time projection coefficients of the non-uniform temperature fields are extracted using the multi-parameter proper orthogonal decomposition (POD) method. The Fully-connected Long Short-Term Memory (FC-LSTM) networks are used to establish the prediction relationship between the measurements and the full-field temperatures. The problem to estimate the projection coefficients of the spatial feature bases is solved by optimizing the loss function using Adam algorithm. The numerical examples show an approximation error less than 5% and the reconstruction time-consuming less than 1 s in 3D temperature field reconstruction (TFR). The proposed reconstruction method is robust at different measurement noise levels and tested point arrangements. [ABSTRACT FROM AUTHOR]

Regarding the nonconvexity of the optimal power flow (OPF) problem, power industry typically employs convex approximation approaches to satisfy the computational demand. However, the accuracy performances of the convex OPF models are hard to be predicted in actual operating power systems. To control the maximum effect of the approximation error on power dispatch, it is important to solve and compare the worst-case approximation errors of these OPF models (i.e., the error bounds). In this paper, the error bound is represented by the maximum gap of the optimal solutions of the nonconvex and convex OPF models (i.e., the optimality gap) which is further equivalent to be the Hausdorff distance of the feasible regions of the two OPF models. Subsequently, an optimization model is proposed to generally solve the Hausdorff distance as well as the error bound. Case studies show that the proposed error bound is tight and effective when the power system is not seriously congested. The relaxation rate of the error bounds is less than 5% when congestion rate is less than 10%. It provides theoretical guidance for selecting the convex OPF model under the condition without historical operating data. • The error bound of convex OPF models is generally modeled based on Fenchel duality. • An approach for solving error bounds is proposed from Hausdorff distance calculation. • The accuracy and application of the error bound measurement are numerically presented. [ABSTRACT FROM AUTHOR]

A novel cascade visual control scheme is proposed to tailor for electrically driven robotic manipulators that operate under kinematic and dynamic uncertainties, utilizing an uncalibrated stationary camera. The proposed control approach incorporates adaptive weight radial basis function neural networks (RBFNNs) to learn the behaviors of the uncertain dynamics of the robot and the joint actuators. The controllers are designed to nullify the approximation error and mitigate unknown disturbances through an integrated robust adaptive mechanism. A major advantage of the proposed approach is that prior knowledge of the dynamics of the robotic manipulator and its actuator is no longer required. The controller autonomously assimilates the robot and actuator dynamics online, thereby obviating the need for fussy regression matrix derivation and advance dynamic measurement to establish the adaptive dynamic parameter update algorithm. The proposed scheme ensures closed-loop system stability, bounded system states, and the convergence of tracking errors to zero. Simulation results, employing a PUMA manipulator as a testbed, substantiate the viability of the proposed control policy. [ABSTRACT FROM AUTHOR]

The demand for thermal power generation from natural gas has increased globally due to its cleaner burning properties compared to other fossil fuels. Optimizing the gas flow through the network to meet this demand is challenging due to the nonconvex Weymouth equation constraining gas flow and nodal pressures in pipelines. Traditional methods for addressing this nonconvexity lead to significant approximation errors or high operational costs. This study poses the Weymouth constraint as a Mathematical Programming with Complementarity Constraints (MPCC) for an optimal gas flow problem. The complementarity constraints reformulate the discontinuous sign function using binary-behaving continuous variables. This MPCC-based approach avoids solving mixed-integer programming problems while enhancing the accuracy of conventional linear and second-order approximations. Testing the approach on various interconnected systems, including Colombia's national gas transportation grid, demonstrated significant reductions in Weymouth approximation errors, thereby supporting effective optimization for interconnected networks. [ABSTRACT FROM AUTHOR]

In the present article, we extend the results of the neural network interpolation operators activated by smooth ramp functions proposed by Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016). We give different results from Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016) we discuss the high-order approximation result using the smoothness of φ and a related Voronovskaya-type asymptotic expansion for the error of approximation. In addition, we showcase the related fractional estimates result and the fractional Voronovskaya type asymptotic expansion. We investigate the approximation degree for the iterated and complex extensions of the aforementioned operators. Finally, we provide numerical examples and graphs to effectively illustrate and validate our results. [ABSTRACT FROM AUTHOR]

This research explored the number of returns to the origin within the framework of a symmetric simple random walk. Our primary objective was to approximate the distribution of return events to the origin by utilizing the half-normal distribution, which is chosen for its appropriateness as a limit distribution for nonnegative values. Employing the Stein’s method in conjunction with concentration inequalities, we derived an exponential non-uniform bound for the approximation error. This bound signifies a significant advancement in contrast to existing bounds, encompassing both the uniform bounds proposed by Döbler [1] and polynomial non-uniform bounds presented by Sama-ae, Chaidee, and Neammanee [2], and Siripraparat and Neammanee [3]. [ABSTRACT FROM AUTHOR]

The Chirp-Fourier transform is one of the most important tools of the modern signal processing. It has been widely used in the fields of ultrasound imaging, parameter estimation, and so on. The key to its application lies in the sampling and fast algorithms. In practical applications, nonuniform sampling can be caused by sampling equipment and other reasons. For the nonuniform sampling, we utilized function approximation and interpolation theory to construct different approximation forms of Chirp-Fourier transform kernel function, and proposed three fast nonuniform Chirp-Fourier transform algorithms. By analyzing the approximation error and the computational complexity of these algorithms, the effectiveness of the proposed algorithms was proved. [ABSTRACT FROM AUTHOR]

Corporate social responsibility (CSR) is a constantly evolving concept that reflects changes in society and the expectations of stakeholders in a process that leads companies to adapt to respond to new challenges and demands. Similarly, the theory and practice of CSR have moved from regulatory compliance to a more holistic approach that requires more sophisticated models that allow for a deeper understanding. Following the fact that companies now have access to a greater amount of information related to their social and environmental performance, as well as comparative data from other organizations, which is why it is necessary to have a management tool that allows collecting, process, and generating more accurate and meaningful metrics that more effectively reflect a company's impact on society. Consequently, the present work aims to identify a system of variables and dimensions representative of the management of organizations that allows the validation of a generic model of corporate social responsibility determined through factorial and structural modeling capable of reflecting more accurate and up-to-date dimensions and metrics of model variables and dimensions. To carry out the study, a sample of 667 middle and senior managers from medium and large companies in Guayas, Ecuador, were randomly contacted in their respective business contexts by trained interviewers. The data processing was carried out in stages. First, the exploratory factorial analysis method of the items was applied to form relevant factors. Then, the dimensions' structural modeling was formulated and ratified through the pertinent goodness-of-fit indices. The results determined a system of correlated factors whose items present estimators of commonality and high and significant factorial loads, excluding variables that did not exceed the sensitization criteria applied. Finally, a model made up of 15 factors and 66 variables ratified using the comparative adjustment indices, the chi-square likelihood ratio, and the mean square error of approximation is introduced, confirming the proposed corporate social responsibility model. [ABSTRACT FROM AUTHOR]

Privacy-preserving neural networks offer a promising solution to train and predict without user privacy leakage, and fully homomorphic encryption (FHE) stands out as one of the key technologies, as it enables homomorphic operations over encrypted data. However, only addition and multiplication homomorphisms are supported by FHE, and thus, it faces huge challenges when implementing non-linear functions with ciphertext inputs. Among the non-linear functions in neural networks, one may refer to the activation function, the argmax function, and maximum pooling. Inspired by using a composition of low-degree minimax polynomials to approximate sign and argmax functions, this study focused on optimizing the homomorphic argmax approximation, where argmax is a mathematical operation that identifies the index of the maximum value within a given set of values. For the method that uses compositions of low-degree minimax polynomials to approximate argmax, in order to further reduce approximation errors and improve computational efficiency, we propose an improved homomorphic argmax approximation algorithm that includes rotation accumulation, tree-structured comparison, normalization, and finalization phases. And then, the proposed homomorphic argmax algorithm was integrated into a neural network structure. Comparative experiments indicate that the network with our proposed argmax algorithm achieved a slight increase in accuracy while significantly reducing the inference latency by 58 % , as the homomorphic sign and rotation operations were rapidly reduced. [ABSTRACT FROM AUTHOR]

Introduction. The problem of approximation can be considered as the basis of computational methods, namely, the approximation of individual functions or classes of functions by functions that are in some sense simpler than the functions being approximated. Most often, the role of an approximant is played by a set of algebraic polynomials or (in the case of a periodic function) a set of trigonometric polynomials of a given order. The ideas and methods of approximation theory are used in various fields of science, especially applied areas, since tasks related to the need to replace one object with another, close in one sense or another to the first, but easier to study, arise very often. The purpose of the paper is consider the problems of approximation of a function, which is given by its values in a certain set of nodal points on a certain interval and belongs to a certain class of functions by trigonometric Fourier series, using the quadrature formulas for calculating integrals of fast oscillating functions on this class of functions, which are optimal in accuracy and close to them. The main attention is paid to the study of the sources of error of the proposed approach to function approximation. Results. Effective approximation algorithms from classes of differentiable functions with the help of Fourier series are proposed, using the Fourier coefficients optimal in accuracy and close to them on the given classes of quadrature formulas for calculating integrals of fast-oscillating functions to determine the Fourier coefficients. The error estimates of the proposed approximation algorithms using the quadrature formulas for calculating the Fourier coefficients of the optimal accuracy and close to them for calculating integrals of fast-oscillating functions from classes of differential functions with given values at the nodes of a fixed grid are presented. The corresponding quadrature formulas and constructive estimates of the error of the method of approximation of functions of these classes are given. Conclusions. Efficient by precision algorithms for approximating functions from classes of differentiable functions by means of Fourier series are constructed using the optimal accuracy and close to them quadrature formulas for calculating integrals of fast-oscillating functions from the above classes of functions to calculate the Fourier coefficients. A comprehensive analysis of the quality of the constructed algorithms for approximating functions by finite sums of the Fourier series is carried out.

Introduction. Function approximation (approximation or restoration) is widely used in data analysis, model building, and forecasting. The goal of function approximation is to find the function that best approximates the original function. This can be useful when the original function is too complex to analyze or when a model needs to be simplified for more efficient computation or interpretation. Function approximation is an important tool in science, engineering, economics, and other fields where data analysis and modeling are required. It allows you to simplify complex functions, identify patterns in the behavior of the object of study, and predict the value of a function beyond the available data. The purpose of the paper is consider the problems of approximation of a function, which on some interval is given by its values in some set of nodal points and belongs to some class of functions by trigonometric Fourier series with a given accuracy and at fulfillment of given constraints on its execution time. The main attention is paid to obtaining estimates of computational complexity (implementation time) and solving the problem of function approximation by Fourier series with a given or maximum possible accuracy using efficient algorithms for solving optimization problems. Results. The general formulation of the problem of approximation of functions by Fourier series in accordance with the technology of solving problems of computational and applied mathematics with specified values of quality characteristics is presented. Estimates of the error of the proposed approximation algorithms using for the computation of Fourier coefficients the optimal in accuracy and close to them quadrature formulas for the computation of integrals from rapidly oscillating functions of the classes of Helder and Lipschitz with given fixed values in the nodes of a fixed grid are given. The corresponding quadrature formulas and constructive estimates of the error of the method of approximation of functions of the specified classes are given. Estimates of computational complexity of the given algorithms are obtained, which allow us to set real constraints on the time of algorithm implementation with a given or maximum possible accuracy. Conclusions. A comprehensive analysis of the quality of the considered algorithms for the approximation of functions by Fourier series using the accuracy-optimal (or close to them) quadrature formulas for the computation of Fourier coefficients for the computation of integrals from rapidly oscillating functions is presented. The estimates of their main characteristics – accuracy and computational complexity – are obtained.

During the actual flight of unmanned helicopters, there are widespread output constraints, system uncertainties and external disturbances, which bring significant threats to their safe flight. To keep the output constraints from violating predefined boundaries, the barrier Lyapunov function (BLF) is utilized in this paper. The system uncertainties are estimated by the fuzzy logic system (FLS). The compound disturbances composed of the external disturbances and the approximation errors caused by FLS are approximated by the fixed-time disturbance observers (FTDOs). According to the analysis of FTDOs and fixed-time stability theory, the observer errors and state errors converge to a region of the origin in fixed-time, and the appropriate parameters of FTDOs and controller are designed to ensure that the disturbance convergence time is faster than the one of the proposed controller, which further improves the stability of the system. Moreover, the convergence time of the FTDOs FTDOs and the proposed approach only depends on the design parameters under any initial conditions with constrained outputs. The performance of the proposed control approach is verified through the simulation results. [ABSTRACT FROM AUTHOR]

Approximate numerical methods are one of the most used strategies to extract information from many-interacting-agents systems. In particular, numerical approximations are of extended use to deal with epidemic, ecological and biological models, since unbiased methods like the Gillespie algorithm can become unpractical due to high CPU time usage required. However, the use of approximations has been debated and there is no clear consensus about whether unbiased methods or biased approach is the best option. In this work, we derive scaling relations for the errors in approximations based on binomial extractions. This finding allows us to build rules to compute the optimal values of both the discretization time and number of realizations needed to compute averages with the biased method with a target precision and minimum CPU-time usage. Furthermore, we also present another rule to discern whether the unbiased method or biased approach is more efficient. Ultimately, we will show that the choice of the method should depend on the desired precision for the estimation of averages. The binomial method, and similar approximations, are often used for numerical simulations of population models in mathematical epidemiology and ecology. The authors study the binomial method to approximate a stochastic dynamic and compare it with an unbiased Gillespie method for stochastic simulation deriving insights on optimal discretization schemes for the binomial method and provide corresponding rules that would indicate if binomial or unbiased solution methods are more efficient. [ABSTRACT FROM AUTHOR]

In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety of activation functions. We introduce two main tools for estimating the metric entropy, approximation rates, and n-widths of these spaces. First, we introduce the notion of a smoothly parameterized dictionary and give upper bounds on the nonlinear approximation rates, metric entropy, and n-widths of their absolute convex hull. The upper bounds depend upon the order of smoothness of the parameterization. This result is applied to dictionaries of ridge functions corresponding to shallow neural networks, and they improve upon existing results in many cases. Next, we provide a method for lower bounding the metric entropy and n-widths of variation spaces which contain certain classes of ridge functions. This result gives sharp lower bounds on the L 2 -approximation rates, metric entropy, and n-widths for variation spaces corresponding to neural networks with a range of important activation functions, including ReLU k activation functions and sigmoidal activation functions with bounded variation. [ABSTRACT FROM AUTHOR]

This article presents an adaptive neural network (ANN) control scheme based on a disturbance observer that can achieve trajectory tracking control of robotic manipulators under external disturbances and dynamic model uncertainties. Firstly, an ANN controller based on full-state feedback is derived using the backstepping technique to achieve an online approximation of uncertainty. The integral sliding mode surface with a position error is introduced into the controller, which reduces the steady-state error of the system and enhances robustness. Then, a novel disturbance observer is designed to estimate both the approximation errors of the ANN and external disturbances, and to provide compensation for the controller, effectively suppressing the trajectory tracking errors caused by approximation errors and disturbances. Subsequently, the Lyapunov stability theory is utilized to demonstrate the stability of the developed control strategy and the boundedness of all closed-loop signals. Finally, numerical simulations are used to confirm the efficacy of the proposed control method. [ABSTRACT FROM AUTHOR]

We deal with multivariate Brass-Stancu-Kantorovich operators depending on a non-negative integer parameter and defined on the space of all Lebesgue integrable functions on a unit hypercube. We prove Lp-approximation and provide estimates for the Lp-norm of the error of approximation in terms of a multivariate averaged modulus of continuity and of the corresponding Lp-modulus. [ABSTRACT FROM AUTHOR]

This paper proposes an analytical three-dimensional propulsion process of electric sail considering constant thrust angle. Due to the small characteristic acceleration of electric sails, its dimensionless form is defined as a perturbation parameter. For an electric sail with a determined initial state, an analytical approximate three-dimensional trajectory consisting of a primary term and perturbation terms can be derived by considering a reasonable approximation in the equation of motion. Furthermore, rectification steps are introduced to minimize approximation error caused by reasonable approximation and continuous changes in elements of osculating orbit. Despite the addition of fixed step rectification, the analytical estimation time is much shorter than the integration time, as the entire approximation process, including the rectification step, is analytical. The time required for analytical approximation is much lower than that for integration. Significantly, the parking orbit of the electric sail required for analytical approximation can be arbitrary. Based on the characteristics, the proposed analytical approximate three-dimensional trajectory is suitable for fast trajectory prediction of various complex deep-space exploration missions, especially for splicing trajectories with different pitch angles to solve problems with fixed targets. [ABSTRACT FROM AUTHOR]

To make better decisions on approximation, we may need to increase reliable and useful information on different aspects of approximation. To enhance information about the quality and certainty of approximating the solution of an Apollonius-type quadratic functional equation, we need to measure both the quality and the certainty of the approximation and the maximum errors. To measure the quality of it, we use fuzzy sets, and to achieve its certainty, we use the probability distribution function. To formulate the above problem, we apply the concept of Z-numbers and introduce a special matrix of the form diag (A , B , C) (named the generalized Z-number) where A is a fuzzy time-stamped set, B is the probability distribution function, and C is a degree of reliability of A that is described as a value of A ∗ B . Using generalized Z-numbers, we define a novel control function to investigate H–U–R stability to approximate the solution of an Apollonius-type quadratic functional equation with quality and certainty of the approximation. [ABSTRACT FROM AUTHOR]

The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project. [ABSTRACT FROM AUTHOR]